Statistical Techniques IIEXST7015Split-plot and Repeated Measures Designs 11223456789101123a_SplitPlot 1Split-plot and Repeated Measure DesignsThe Split-plot and Repeated Measures "Designs" combine elements of design (error structure) and treatment arrangement concepts. These are designs with two levels, a "Main Plot", with its own treatment and error, and a "Sub-plot", with its own treatment and error. It is possible to have more than just an "a priori" treatment in both levels.23a_SplitPlot 2Split-plot Designs (continued)The (minimum of) two treatments (from the main and sub plots) are usually cross classified . Either Main or Subplot may have nested error structure. The simplest split plot would have the following model (CRD). Yijk= µ+τ1i+γij+τ2k+τ1τ2ik+ εijk23a_SplitPlot 3Split-plot Designs (continued)CAABBAABCCABB CCExample with CRD main plot. 23a_SplitPlot 4Split-plot Designs (continued)Each plot SPLIT for a new treatment.CAABBAABCCABB CCFGFGGFFGGFGFFGFGGFFGGFGFFGFGFG23a_SplitPlot 5Split-plot Designs (continued)Source d.f.Treatment1 t1-1 = 2Error(a) t1(n-1) = 12Treatment 2 t2-1 = 1Tmt1*Tmt2 (t1-1)(t2-1) = 2Error(b) t1*(t2-1)(n-1) = 12Total t1*t2*n-1 = 29Split-plot design source table. The d.f. for error(b) is the usual t1*t2*(n-1) less the d.f. for error(a), t1*(n-1), giving t1*(t2-1)(n-1). 23a_SplitPlot 6Split-plot Designs (continued)Split-plot design - examples of splitsWe may split a plot to do a new treatment, e.g. an agricultural experiment with fertilizer treatments in plots may have a herbicide applied to half of each plot an not to the other half. A soil study of contaminants may measure levels of the chemical of interest at various levels in a soil core (0-5 cm, 6-10 cm, 11-15 cm, etc), so the core is split. 23a_SplitPlot 7Split-plot Designs (continued)Split-plot split examples (continued ) A study of the growth of plants, e.g. Spartina in a marsh, may split the plant into above ground, root and rhizome biomass. Anytime a treatment occurs within an experimental unit, we have a split-plot. If we are studying diets of fish, and put a male and female fish in each aquaria, weight gain of hogs with large and small hogs in each pen, etc. 23a_SplitPlot 8Split-plot Designs (continued)More complex designs are possible. The main plot may be an RBD, the main plot and/or sub plot treatments may be factorial or nested. It is possible to have plots that are split twice, or split and measured repeatedly. These designs are complicated, difficult to analyze and difficult to interpret. So why do you do them? 23a_SplitPlot 9Split-plot Designs (continued)Split plot design with an RBD main plot. AAAABBBCCCCBdfegdfegdfegdfegdfegdfegdfegdfegdfegdfegdfegdfeg23a_SplitPlot 10Split-plot Designs (continued)This design has two blocks, three levels in the main plot treatment and four levels in the subplot treatment. For the main plot the analysis is the same as any RBD. This one will have treatments, blocks, treatment*block interaction and replicated experimental units in blocks. Yijk=µ+βi+τ1j+τβij+γijk+τ2l+τ1τ2il+τ1τ2βijl+εijkl23a_SplitPlot 11Split-plot Designs (continued)Source d.f.Block b-1 = 1Treatment 1 t1-1 = 2B*T1(b-1)(t1-1) = 2Error(a) bt1(n-1) = 6Treatment 2 t2-1 = 3Tmt1*Tmt2 (t1-1)(t2-1) = 6B*T2+B*T1*T2(pooled)(b-1)(t1-1)(t2-1) = 9Error(b)b*t1*(t2-1)(n-1) = 18Total b*t1*t2*n-1 = 47Source table RBD main plot in split-plot. 23a_SplitPlot 12Repeated measuresThe repeated measures design is similar to a split-plot. We have a "main plot", which can be any of the designs we have discussed previously (CRD, RBD, LSD). We then take repeated measurements over time within the plots. If these "repeated measures" are independent, then this "time" factor is just cross-classified with the treatment. 23a_SplitPlot 13Repeated (continued)If, however, the measurements are NOT independent, we have a repeated measures design. Independence? Again? Yep. What do I mean by independent? For example, if you are sampling sugar content of an ear of corn from a plot, or the height of Spartina in a plot, you ask, "are they independent or not?" 23a_SplitPlot 14Repeated (continued)If you measure a different ear of corn from a different plant each time, or measure a different Spartina plant, they are probably independent. However, if you measure a kernel from the same ear of corn, or the same Spartina plant each time, they are NOT independent. 23a_SplitPlot 15Repeated (continued)Some examples of split plot and repeated measures variables. Pre-post tests on people, in fact most any experiment where several levels of a treatment(s) are measured on the same subject (= a person). Soil samples or water samples at different depths (in the same site). Epiphytes on Spartina counted below, at and above the tide line (on the same plant).23a_SplitPlot 16Repeated (continued)More examples Studies on plants like sugar cane where we measure production in year1, year2 and year3 on the same biological material. Ditto for asparagus, trees, artichokes, etc. In general, any time your experimental unit has a treatment applied to each experimental unit, this is a split plot. If the experimental unit is measure over time it is repeated measures. 23a_SplitPlot 17Repeated (continued)Why is this independence important? What can we do about it? Lets BRIEFLY revisit the X and X'X matrices. The X matrix for designs consists of columns of 0 values and 1 values, arranged to distinguish between categories. 23a_SplitPlot 18Repeated (continued)For a simple CRD with 4 treatment levels the X'X matrix may look like the following. X = 1 0 0 01 0 0 00 1 0 00 1 0 00 0 1 00 0 1 00 0 0 10 0 0 123a_SplitPlot 19Repeated (continued)For a simple CRD with 4 treatment levels the X'X matrix would look like the following. X'X =n10 0 00 n20 00 0 n300 0 0 n423a_SplitPlot 20Repeated (continued)For a simple CRD with 4 treatment levels the (X'X)-1matrix would look like the following. (X'X)-1=1/n10 0 00 1/n20 00 0 1/n300 0 0 1/n423a_SplitPlot 21Repeated (continued)To get the variances and covariances we multiply by the MSE, as you know. This gives MSE/n on the main diagonal, and zeros on the off diagonal. All those zeros on the off diagonal mean that THERE IS NO COVARIANCE BETWEEN THE TREATMENTS. This is well and good, we do not expect covariances between the independently sampled treatments. 23a_SplitPlot 22Repeated (continued)But for the split plot and repeated